THE QUADRATRIX OF HIPPIAS
229
1 Pappus, iv, pp. 252. 26-254. 22.
flourished about 350 B.C., that is to say, some time before
Euclid, it is worth while to note certain propositions which
are assumed as known. These are, in addition to the theorem
of Eucl. VI. 33, the following: (1) the circumferences of
circles are as their respective radii ; (2) any arc of a circle
is greater than the chord subtending it; (3) any arc of a
circle less than a quadrant is less than the portion of the
tangent at one extremity of the arc cut off by the radius
passing through the other extremity. (2) and (3) are of
course equivalent to the facts that, if a be the circular measure
of an angle less than a right angle, sin a < a. < tan a.]
Even now we have only rectified the circle. To square it
we have to use the proposition (1) in Archimedes’s Measure
ment of a Circle, to the effect that the area of a circle is equal
to that of a right-angled triangle in which the perpendicular
is equal to the radius, and the base to the circumference,
of the circle. This proposition is proved by the method of
exhaustion and may have been known to Dinostratus, who
was later than Eudoxus, if not to Hippias.
The criticisms of Sporus, 1 2 in which Pappus concurs, are
worth quoting :
(1) ‘The very thing for which the construction is thought
to serve is actually assumed in the hypothesis. For how is it
possible, with two points starting from B, to make one of
them move along a straight line to A and the other along
a circumference to D in an equal time, unless you first know
the ratio of the straight line AB to the circumference BED ?
In fact this ratio must also be that of the speeds of motion.
For, if you employ speeds not definitely adjusted (to this
ratio), how can you make the motions end at the same
moment, unless this should sometime happen by pure chance 1
Is not the thing thus shown to be absurd 1
(2) ‘ Again, the extremity of the curve which they employ
for squaring the circle, I mean the point in which the curve
cuts the straight line AD, is not found at all. For if, in the
figure, the straight lines CB, BA are made to end their motion
together, they will then coincide with AD itself and will not
cut one another any more. In fact they cease to intersect
before they coincide with AD, and yet it was the intersection
of these lines which was supposed to give the extremity of the